[J. Res. Natl. Inst. Stand. Technol. 106 Quasicrystals · Quasicrystals Volume 106 Number 6 November–December 2001 John W. Cahn National Institute of Standards and Technology, Gaithersburg,

Volume 106, Number 6, November–December 2001Journal of Research of the National Institute of Standards and Technology

[J. Res. Natl. Inst. Stand. Technol. 106, 975–982 (2001)]

Quasicrystals

Volume 106 Number 6 November–December 2001

John W. Cahn

National Institute of Standards andTechnology,Gaithersburg, MD 20899-8555

[email protected]

The discretely diffracting aperiodic crystalstermed quasicrystals, discovered at NBSin the early 1980s, have led to much inter-disciplinary activity involving mainlymaterials science, physics, mathematics,and crystallography. It led to a new un-derstanding of how atoms can arrangethemselves, the role of periodicity in na-ture, and has created a new branch of crys-tallography.

Key words: aperiodic crystals; newbranch of crystallography; quasicrystals.

Accepted: August 22, 2001

Available online: http://www.nist.gov/jres

1. Introduction

The discovery of quasicrystals at NBS in the early1980s was a surprise [1]. By rapid solidification we hadmade a solid that was discretely diffracting like a peri-odic crystal, but with icosahedral symmetry. It had longbeen known that icosahedral symmetry is not allowedfor a periodic object [2].

Periodic solids give discrete diffraction, but we didnot know then that certain kinds of aperiodic objects canalso give discrete diffraction; these objects conform to amathematical concept called almost-1 or quasi-periodic-ity [3]. By definition all quasi-periodic objects diffractdiscretely, even though they are not periodic. Quasiperi-odic objects can have any of the infinite set of pointgroup symmetries listed as non-crystallographic in theInternational Tables for Crystallography [4]; becausethey have a single rotation axis of order 5, or one greaterthan or equal to 7, or have icosahedral symmetry with itssix intersecting 5-fold axes.

1 For reason discussed below we need not be concerned with almostperiodicity.

Crystal periodicity has been an enormously importantconcept in the development of crystallography. Hauy’shypothesis that crystals were periodic structures led togreat advances in mathematical and experimental crys-tallography in the 19th century. The foundation of crys-tallography in the early nineteenth century was based onthe restrictions that periodicity imposes. Periodic struc-tures in two or three dimensions can only have 1,2,3,4,and 6 fold symmetry axes. With no exceptions, eachcrystal was found to conform to one of only 32 ways ofcombining these symmetry axes, the so-called“crystallographic” point group symmetries. Externalforms of periodic crystals were found to be limited tocombinations of only 47 forms (32 general and 15 spe-cial) made of symmetrically arranged bounding planes[5]. Cubes, octahedra, and tetrahedra, for instance, areexamples of special forms belonging to the cubic pointgroups, octahedra to point groups 432, m3, and m3m ,tetrahedra to 23 and 43m , and cubes to all five. In thenineteenth century each known crystal could be fit intoone (or more) of these 32 point groups by the examina-tion of its external form. That no additional

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form was found could be taken as proof that all crystalsare periodic. Regular icosahedra and dodecahedra arespecial forms of both icosahedral point groups, 235 andm35. All icosahedral forms have fifteen intersecting 2-and ten intersecting 3-fold axes, as well as six intersect-ing non-crystallographic 5-fold axes.

With the assumption of periodicity, the mathematicalaspects of crystallography were set and completelyworked out in the 19th century; that aspect became analmost closed field. In two and three dimensions thenumber of crystal systems, point groups, and plane orspace groups were all enumerated. When the allowedsymmetry axes are combined with translations, it wasshown that there are only 230 space groups in threedimensions. In two dimensions there are only ten pointgroups and seventeen plane groups. An elementaryproof why this listing contains every case allowed byperiodicity and why no others are allowed has long beenavailable in popular mathematics books [6]. Such com-plete listing are called catalogs. Each one of the seven-teen are beautifully illustrated in etchings by M. Escher[7], as well as Moorish tilings and Turkish carpets.Extensions were developed for color groups and forcrystallography in higher dimensions. Magnetic struc-tures and their 1609 Shubnikov space groups are anexample of such an extension in which spins, up ordown (or two colors), are treated as if in a fourth dimen-sion [8].

With the advent of x-ray diffraction in 1912, externalform became less important. Crystals became defined asperiodic arrangements of identical unit cells. The domi-nant work of crystallographers became structure deter-minations by diffraction to find the atom content of oneunit cell. The method depends on an assumed periodic-ity, and the results usually confirmed it.

2. Discussion

Had we found a crystal? Many definitions of crystalsare in use, some have changed over the centuries. Oursolid was metallic and thus not a “clear transparentmineral.” It can be grown to form “convex solids en-closed by symmetrically arranged plane surfaces, inter-secting at definite and characteristic angles.” Accordingto the latter of these older definitions, quasicrystals arecrystals. The discovery in 1912 that crystals could dif-fract x-rays discretely implied either their periodicity orquasiperiodicity. But as noted above, the subsequentstructure determinations, seem to have led to the accep-tance of a definition of crystals based on the periodicityof their internal structure, and one which unnecessarilyruled out quasiperiodicity. But by 1992 the IUCr Ad

Interim Commission on Aperiodic Crystals wrote “by‘crystal’ we mean any solid having an essentially dis-crete diffraction pattern, and by ‘aperiodic crystal’ wemean any crystal in which three-dimensional lattice pe-riodicity can be considered to be absent” [9]. By thislatest definition, our solid is a crystal, albeit an aperiodicone. It is a “quasiperiodic crystal” or quasicrystal forshort, a word coined by Levine and Steinhardt [10].

Our surprising discovery created quite a stir and hasinfluenced research in many fields, not just crystallogra-phy, but also materials science, physics, mathematics[11,12], biology [13,14], and even art. There have beenabout 10 000 papers in these fields and many conferenceproceedings [15]. Hundreds of quasicrystals have beenfound since, some with non-crystallographic axial sym-metries, pentagonal, octagonal [16], decagonal [17],and dodecagonal [18]. The crystals with axial sym-metries are usually periodic along the symmetry axis,and quasiperiodic in the basal plane.

Quasiperiodicity is a form of aperiodicity that hasmany of the attributes of periodicity. As one of theirdefining properties, Fourier transforms of quasiperiodicfunctions are discrete sets of delta-functions; they canalways be expressed as a series of sine and cosine terms,but with incommensurate lengths, or a number of arith-metically independent basis vectors that exceeds thenumber of independent variables. Physically, aquasiperiodic object diffracts to give a pattern withsharp Bragg spots. But whereas diffraction from a peri-odic object forms a reciprocal lattice that can be indexedwith a set of d reciprocal basis vectors, where d is thedimension, the diffraction pattern from a quasiperiodicobject requires a finite number, D > d , independent ba-sis vectors. An important consequence of this is that anyquasiperiodic function can always be represented as aperiodic function in D dimensions. The aperiodic func-tion then is a d -dimensional cut of this periodic function.If D is infinite, the function is called almost periodic. Wehave so far not been concerned with almost periodicity,since in any experiment D is less than or equal to thenumber of observed reflections, and thus is finite.

As a simple example consider the one-dimensionalfunction f (x ) = cos x + cos bx . The Fourier transformconsists of two delta functions. If b is rational, f isperiodic, the two delta functions can be indexed with asingle reciprocal lattice vector. If b is irrational, f isquasiperiodic; there are two incommensurate lengths inthe Fourier transform; D = 2. The function f (x ,y ) = cosx + cos y is periodic in two dimensions; the quasiperi-odic one-dimensional f (x ) is recovered by settingy = bx . Note that there would be no diffuse scatteringfrom a quasiperiodic object with f as its density func-tion.

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Figure 1 shows the first diffraction pattern taken froma quasicrystal oriented along the 5. Note first the discretediffraction and the apparent 10-fold symmetry. Note thatthere are no systematic rows; spots twice or three timesas far as a bright spot are much weaker if seen at all.Note that the ratio of distances in any row is the “goldenmean” � , (� = 2 cos 360 = (1 + �5)/2 = 1.618034...),and that � occurs naturally in the ratios of the magni-tudes of vector sums of spots at 360 from one another.Lastly note that it is impossible to index this pattern withjust three reciprocal lattice vectors.

Our brains often take us to higher dimensions to sim-plify what is seen. Every triplet of rhombs meeting at atriple corner in the Penrose tiling in Fig. 2 can look likea three-dimensional cube, but they are arrayed in severalorientations, and the same rhomb can seem to have dif-ferent orientations depending on which other two neigh-boring rhombs it is grouped with. In five dimensions thistiling finally becomes simple and unambiguous witheach edge along a specific one of five orthogonal axes

and each rhomb becomes a square with a unique orien-tation. A zigzag path along the lines of the tiling be-comes a Cartesian path in five dimensions, and a five-index coordinate system specifies each corner. In fivedimensions the Penrose tiling is confined to the set of allthe lattice points within a slice bounded by two parallelplane hypersurfaces with irrational orientations.

Since the (111) plane of the primitive cubic lattice isthe two-dimensional hexagonal lattice, the three-dimen-sional hexagonal lattice can be considered as the (1110)plane of a four-dimensional cubic lattice [19]. This ra-tional cut can simplify the understanding of indexinghexagonal structures. The 4-index specification of apoint �hkil� in a four-dimensional cubic structure can beused to specify a point in the real three-dimensionalhexagonal crystal. For the point �hkil� to be in the three-dimensional crystal, it has to be on the (1110) hyper-plane of the four-dimensional cubic structure, i.e., it hasto have h + k + i = 0. Distances between two such pointscan be computed more easily in the 4-index notation.

Fig. 1. The first electron diffraction pattern from a quasicrystal [1]. Note the forbid-den, 10-fold axis, the absence of systematic rows, and the need for more than threevectors to index all the spots.

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Fig. 2. Penrose tilings are quasiperiodic. Groups of three tiles around a trivalent point look likethree-dimensional cubes with 90� between line segments, but the orientation of some tiles isambiguous. In five dimensions this ambiguity is removed, all line segments can be orthogonal, anthen this entire pattern will fit between two parallel hyperplanes.

A physical example of a two-dimensional quasiperi-odic object is the surface obtained by cutting a three-di-mensional crystal by an irrational plane. In this examplethe three basis vectors of the periodic three-dimensionalcrystal are needed to describe this two-dimensionalquasiperiodic surface. Because the cutting plane is irra-tional the surface cannot be periodic; it will never gothrough exactly the same point in two different unitcells. Yet when the plane comes close to the same pointin some distant unit cell, another plane through thatpoint will be very close all the way out to infinity. Theaperiodic structures these planes represent will superim-pose with little error all the way to infinity. That distancebetween the points is an approximate translation vector,whose existence depends on the specification of howsmall a superposition error we require. For a periodicfunction the superposition would be exact; the transla-tion can be repeated indefinitely, and thus form a lattice.For a quasiperiodic function, repetition of any transla-tion increases the mismatch, and eventually the errorbecomes too large; thus the translations in a quasiperi-odic structure do not form a lattice, but what is called a

quasilattice. But the existence of these translations is animportant property of quasiperiodic functions and ofquasicrystals.

There are many kinds of defects in periodic structuresthat have their analogs in quasiperiodic structures. Let usbegin by examining how defects in a three-dimensionalperiodic crystal would appear on a two-dimensionalaperiodic surface. Consider, for example, a metalliccrystal with a CsCl ordering of a body centered cubicstructure to a Pm3m space group with differing occupa-tion of corners and body centers. Such metallic crystalscommonly have internal boundaries, called antiphaseboundaries, separating domains in which the site occu-pations are reversed. Such boundaries break the transla-tional symmetry in an otherwise periodic crystal. Nowconsider a cut of such a crystal on an irrational plane.Although this cut surface is aperiodic, the domainboundary, a translation defect, would clearly be visiblein the quasiperiodic surface. Dislocation lines in threedimensions, intersecting the surface, would show up inthe surface as points with associated Burgers vectors.Since, apart from some small strains, the three-dimen-

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sional structure is perfect away from the dislocation, sois the quasiperiodic surface. Thus we can detect transla-tional faults and imperfections in quasiperiodic objects.Defects in quasicrystals can be understood as defects ina higher-dimensional periodic crystal. Ordering can oc-cur in icosahedral quasicrystals, giving rise to antiphaseboundaries that are five-dimensional hypersurfaces inthe six-dimensional crystal and seen as surfaces in thethree-dimensional quasicrystal [20]. This boundary canalso be seen in an imperfect ordering of a Penrose tilingin which adjacent corners alternate black and white.Dislocation lines in icosahedral quasicrystals arise froma four-dimensional defect surface in six dimensions.Mechanical deformation of quasicrystals is a most inter-esting subject. Away from the dislocation line, the qua-sicrystal is perfect, as it would be with dislocations inperiodic crystals.

Although no new symmetry axes appear in goingfrom two to three dimensions, higher dimensions allownew symmetries to be consistent with periodicity. Forthe axial groups a n -fold symmetry axis first becomespossible with translational symmetry if the dimensional-ity equals the totient of n , which is the number of posi-tive integers less than or equal to n which are relativelyprime (no common factors) to n [21]. This is readilyillustrated for any prime number N , whose totient isN�1. Since the (11...1) hyperplane in an N -dimensionalisometric lattice has an N -fold axis and the dimension ofthat plane is N�1, the rule works for all primes. Twohas a totient of 1; three, four, and six have totients of 2;none have 3; five, eight, ten, and twelve have 4, etc. Thusfive, eight, ten, and twelve-fold rotations first appear infour-dimensional periodicity. Icosahedral symmetrywith its intersecting five-fold axes requires six dimen-sions. Each of the six axes in an isometric six-dimen-sional lattice meets the five others at right angles, givingrise to six 10-fold axes. Because there is no point groupwith more than one 10-fold axes in three dimensions, thecuts by irrational planes can only preserve the six 5-fold(or the six 5-fold inversion) axes of the icosahedralsymmetry.

The study of quasicrystals benefited greatly fromprior research in the mathematical subjects of quasiperi-odic functions, aperiodic tilings, and hyperspace crystal-lography. The latter had already been applied in thestudy of modulated crystals [22]. Modulated structureshad been found long before the discovery of quasicrys-tals and had provided some well-documented and under-stood exceptions to periodic crystals. Because theycould be considered as small incommensurate distor-tions of periodic structures with a crystallographic pointgroup, they could be fit into the schemes of crystallogra-phy. But the incorporation of the modulation wavelengthas an additional length provided an impetus for the

development of hyperspace crystallography which al-lowed a periodic indexing in the higher dimension.Modulated structures could then be treated as cuts onirrational planes, and sometimes as projections of aslice, of a four or higher dimensional periodic structure.In an ideal modulated structure, each spot, including thesatellites, is a Bragg peak, indexed with more than three,usually four, numbers.

Consider an icosahedral structure to be an irrationalcut of a six-dimensional cubic structure with a singlelattice parameter. Indexing requires six numbers, whichis obvious in six dimensions, but is also true for three.In a three-dimensional indexing using three orthogonalaxes in a Cartesian system, two indexes are requiredalong each axis, and six number specify each spot,(h + h'� , k + k'� , 1 + 1'� ) [23]. Indexing of icosahedralpowder patterns is also straightforward; the ambiguitiesresulting from superpositions (such as (330) and (411)in bcc powder patterns) are infrequent. After a latticeparameter has been selected, indexing for all six num-bers for single crystals is unambiguous in either three orsix dimensions. Using synchrotron radiation from a sin-gle AlCuFe quasicrystal, Moss and coworkers have mea-sured intensities of about 1200 crystallographically dis-tinct peaks, every peak found using a single icosahedral(quasi)lattice parameter and a six-parameter icosahedralindexing [24].

Structure determinations would seem like a hopelesstask. Has one to describe the structure of an aperiodicsolid out to infinity? Because there is periodicity in thehigher dimensions, one needs only to describe the con-tent of one unit cell (hypercell) in the higher dimen-sional space. Structure determination in six dimensionsis not very different from what it is in three. Once thediffraction peaks from single crystals (or lines frompowders) have been indexed in six dimensions, or inthree with six basis vectors, and their corrected intensi-ties measured, the diffraction pattern can be consideredeither on a three-dimensional reciprocal quasilattice oron a six-dimensional periodic lattice. They are com-pletely equivalent to another, but standard methods ofcrystallographic structure determination for periodicstructures are applicable with little modification to thesix-dimensional data.

Indexing allows Patterson functions to be directly ob-tainable in three or six dimensions from powder data.They have the directional information lost in a radialdistribution function. Although the three-dimensionalPatterson functions are aperiodic and complicated withmany peaks, near the origin they bear striking similari-ties to Patterson functions of related periodic approxi-mants, large cell periodic crystals with compositionsslightly different from quasicrystals [25]. Thus the localatom packing of quasicrystals are found to be very

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similar to that of corresponding periodic phases. Patter-son functions in six dimensions are usually found to bemuch simpler, with only a few peaks in each unit cell.Actual structure determinations have now been carriedout for several quasicrystals with very good residuals[26,27]. Atom positions are described in the six-dimen-sional unit cell by three-dimensional surfaces; the inter-sections of these surfaces periodically repeated in sixdimensions by the irrational three-dimensional plane arethe points that describe the atom positions in the three-dimensions of quasiperiodic structures. In analogy withthe finding of three dimensional periodic structures byfitting balls with atomic radii together, plausiblequasiperiodic structures have been constructed by fittingatomic surfaces into six dimensional unit cells [28].Another technique exploits the known structures of peri-odic approximants to convert the structure determina-tion of the related quasicrystals to the standard crystallo-graphic structure refinement problem [29].

Periodic crystals can be considered a tiling of unitcells, each decorated with atoms. Tilings with noncrys-tallographic symmetries occur in art where the mathe-maticians’ rules about having a limited number of con-gruent tiles and leaving no gaps need not be met. Thediscovery by mathematicians of aperiodic tilings pre-ceded that of quasicrystals. Penrose’s tilings with 5-foldsymmetry seem particularly pertinent; they arequasiperiodic and their diffraction pattern is strikinglysimilar to that of the 5-fold zone of icosahedral qua-sicrystals [30]. By analogy some of the early modelswere based on atomic decorations of three-dimensionalversions of Penrose tiles as if each of the different tileshad the same decoration of filled atom positions.

Three-dimensional structures that give sharp diffrac-tion are either periodic, if the indexing requires threebasis vectors, quasiperiodic, if the indexing requires afinite number, more than three, and almost periodic, ifthe indexing requires an infinite number. All the mathe-matical interest had been with almost periodicity; anyquasiperiodic structure is periodic in a higher dimen-sion.

There was considerable initial resistance to quasicrys-tals. My own initial reaction was that we were seeing aquintuple twin, often seen in cubic crystal, but that waseasily ruled out with data presented in our paper.The angles between the (111) twinning planes(arccos (1/3) � 70.53�) in cubic crystals are sufficientlyclose to 72� that five wedge shaped periodic cubic crys-tal can fill space with some easily detected strain orextra material to fill the missing 7.36�. Twins lead to asuperpositioning of five reciprocal lattices, each givingsystematic rows of periodically spaced diffraction spots.The absence of such systematic rows argues againsttwinning. The other possibility was a very large unit cell

with a structure that will give the strange extinctions toconform to the lack of systematic rows of spots that wenow know is a characteristic of diffraction from qua-sicrystals. Assuming we had a periodic low-symmetrycrystal, we searched unsuccessfully to fit the data withcell constants up 2.5 nm. Even though either the twin-ning or the large unit cell were plausible alternate expla-nations, Linus Pauling became one of the vocal oppo-nents by proposing a double-kill, both a large unit celland what he called icosatwinning. His initial structure,based on his often successful method of fitting atomstogether, had a face centered cubic unit cell with alattice constant of 2.67 nm, containing 1168 atoms (292atoms per primitive cell). His claim [31] to fit our pow-der data led him to write that there was only 1 chancein 10 000 that this unit cell could be wrong, but heignored that his indexing could not fit our publishedsingle crystal pattern. A few years later, he found itnecessary to propose another cell, this time a primitivecubic structure with a lattice constant of 2.34 nm, con-taining 820 atoms [32]. Either of his structures wouldqualify as an approximant, but to the best of my knowl-edge, no one has yet reported finding either of them.

Quasicrystals provided win-win opportunities forcrystallographers: If we were mistaken about them, ex-pert crystallographers could debunk us; if we were right,here was an opportunity to be a trail blazer. While manycrystallographers worldwide availed themselves of theopportunity, U.S. crystallographers avoided it, to a largeextent because of Pauling’s influence. The demonstra-tion by E. Prince that tilings with five-fold symmetrywould give discrete diffraction pattern was a notableexception [33].

The systems that form quasicrystals additionally oftengive periodic crystals with large unit cells, called peri-odic approximants; sometimes there is even a sequenceof approximants with ever larger cell constants [34]. TheFrank-Kasper phases [35] turned out to be examples ofperiodic approximants to quasicrystals that were foundlater. Because their diffraction spots are periodicallyarrayed, approximants are easily distinguished from qua-sicrystals. Quintuple twinning of approximants is some-times seen, as is one case of triple twinning of a qua-sicrystal, giving an apparent 30-fold diffraction pattern[36].

Much has been written about why quasicrystals exist.Although it could not be proven, it was taken as plausi-ble by many eminent scholars that the lowest energyconfiguration of a set of identical atoms or moleculeswould be periodic. Similarly it was assumed that thelowest energy configuration of any mixture of atoms ormolecules would be a periodic arrangement of identicalunit cells forming a stoichiometric compound, or a mix-ture of such periodic structures. Radin has shown quite

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the opposite; for almost any assumed interaction be-tween molecular units, the lowest energy is a quasiperi-odic rather than a periodic structure [37]. He has raisedthe question about whether periodic crystals exist be-cause kinetics are too slow to reach the lowest energystate, or whether there is something special about theinteractions obtained from quantum mechanics.

To this date all quasicrystals have been metallic. Inmetallic structures interatomic distances are deter-mined, but bond angles do not seem to matter. Localatomic configurations thus obtained often do not packwell into periodic structures. Even the simplest one-component metallic structures seem to favor regular te-trahedral arrangements that do not fill space. Whatother local configuration is needed to fill the gaps, anddoes that lead to the orientational order seen in qua-sicrystals and periodicity or quasiperiodicity? Struc-tures are determined by a trade-off between low energylocal packing and the occasional higher energy configu-ration that is geometrically necessary. In order to have aperiodic space-filling arrangement, both the face-cen-tered and hexagonal close packed structures, for exam-ple, introduce the octahedra, a configuration which oneexpects to have a higher energy. The stable quasicrystalsand the approximants are made of two [38] or morechemical components, allowing irregular tetrahedra thathave a better chance of filling space. Whether the ad-justments happen to lead to a periodic approximant or toa quasicrystal often seems to hinge on small changes incomposition or temperature.

While periodic and quasiperiodic structures alwaysgive discrete diffraction, what other kinds of aperiodicstructures diffract [39]? Mathematicians have found averitable “zoo” of orderly dispositions of points in space[11]. Have any of the many that are not quasiperiodicbeen found in nature or made in the laboratory? Anisotropic solid structure was found at NIST in a four-component system in which quasicrystals exist at some-what different compositions. But while metallic glassesusually result from any remaining melt that is cooled toorapidly to crystallize, this solid grew first as if it were acrystal, with an interface and at a composition differentfrom the melt [40]. On continued cooling the melt crys-tallized around this solid. Is this a physical realization ofone of the many other orderly, arrangements of atoms,discussed by mathematicians that is not quasiperiodic?

Lattices are considered important factors in manyphysical problems. For a long time a three-dimensionalcoincidence site lattice was deemed so important thatlaws of twinning were based on the existence of a peri-odic arrangement of a fraction S of lattice sites of bothtwins, even if the lattice sites are not occupied by atoms.Coincidence sites are still considered important in thetheory of grain boundaries, except that the twinner’s S

has become a � . But why should such a three-dimen-sional lattice be so important? The energy surely de-pends only on the fit of atoms at the twin interface orgrain boundary. An extreme case is a merohedral twinfor which S = 1 in which the lattice is continuousthrough the twin boundary. These twins occur in caseswhere the motif has less symmetry than the lattice; thetwin is formed when the motif is rotated by a symmetryoperation of the lattice but not of the motif. We foundthe opposite case in an arrangement of several approxi-mant crystals [41]. Here the icosahedral motif has ap-proximate symmetry operations that are not present inthe cubic lattice. Quasitwins occur when the lattice ro-tates by 72� about an irrational �1� 0� axis while themotif retains its orientation across the grain boundary.The motif has long range orientational order across theboundary as it does in quasicrystals, surely for energyreasons.

Acknowledgments

It has been an interesting 20 years. I am extremelygrateful to many of my colleagues at NIST, at CECM-CNRS, Vitry, France, and at the Technion, Haifa, Israel,but Dan Shechtman, who discovered the Al-Mn qua-sicrystal at NIST, L. Bendersky, who identified the firstdecagonal quasicrystals, and D. Gratias, who taught memuch that was new about crystallography, deserve to besingled out. The rapid solidification work was sup-ported by DARPA; later we benefitted greatly from aPICS agreement between NIST and CNRS.

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837 (1989).

About the author: John W. Cahn is a Senior Fellow inthe NIST Materials Science and Engineering Labora-tory. His research interests include quasicrystals, phasetransitions, metals, microstructure evolution, and crys-tal growth. The National Institute of Standards andTechnology is an agency of the Technology Administra-tion, U.S. Department of Commerce.

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Documents

[J. Res. Natl. Inst. Stand. Technol. 106 Quasicrystals · Quasicrystals Volume 106 Number 6 November–December 2001 John W. Cahn National Institute of Standards and Technology, Gaithersburg,